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$$
  \begin{bmatrix}r_2 \\ V_2 \end{bmatrix} =
  \begin{bmatrix}
    A & B \\
    C & D 
  \end{bmatrix}
  \begin{bmatrix}r_1 \\ V_1 \end{bmatrix}
  
  \;\;\Rightarrow\;\;
  
  \begin{Bmatrix}
    r_2 = A r_1 + B V_1 \\
    V_2 = C r_1 + D V_1
  \end{Bmatrix}
$$

$$ \gamma = \sqrt \frac {n_2} {n_0} $$

$$
  \begin{bmatrix}r_2 \\ V_2 \end{bmatrix} =
  \begin{bmatrix}
    \cos(\gamma L) & \cfrac {\sin(\gamma L)} {n_0 \gamma} \\
    -n_0 \gamma \: \sin(\gamma L) & \cos(\gamma L) 
  \end{bmatrix}
  \begin{bmatrix}r_1 \\ V_1 \end{bmatrix}
$$

$$ V_1 = 0 $$    
$$ r_2 = r_1 \cos(\gamma L) $$
$$ V_2 = -r_1 n_0 \gamma \sin(\gamma L) $$

$$
  \begin{bmatrix}r_3 \\ V_3 \end{bmatrix} =
  \begin{bmatrix}
    1 & F \\
    0 & 1 
  \end{bmatrix}
  \begin{bmatrix}r_2 \\ V_2 \end{bmatrix}
$$

$$ r_3 = r_2 + F V_2 = 0 $$
$$ V_3 = V_2 $$

$$
F = \frac 1 {n_0 \gamma} \frac {\cos(\gamma L)}{\sin(\gamma L)}
$$

$$
F = \frac 1 {n_0 \gamma \tan(\gamma L)}
$$

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